Optimal. Leaf size=24 \[ \text {Int}\left (\frac {1}{(c+d x) (a-a \cos (e+f x))},x\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c+d x) (a-a \cos (e+f x))} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(c+d x) (a-a \cos (e+f x))} \, dx &=\int \frac {1}{(c+d x) (a-a \cos (e+f x))} \, dx\\ \end {align*}
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Mathematica [A] time = 2.40, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) (a-a \cos (e+f x))} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a d x + a c - {\left (a d x + a c\right )} \cos \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (d x + c\right )} {\left (a \cos \left (f x + e\right ) - a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right ) \left (a -a \cos \left (f x +e \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (\frac {{\left (a d^{2} f x + a c d f + {\left (a d^{2} f x + a c d f\right )} \cos \left (f x + e\right )^{2} + {\left (a d^{2} f x + a c d f\right )} \sin \left (f x + e\right )^{2} - 2 \, {\left (a d^{2} f x + a c d f\right )} \cos \left (f x + e\right )\right )} \int \frac {\sin \left (f x + e\right )}{{\left (d x + c\right )}^{2} {\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )}}\,{d x}}{a f} + \sin \left (f x + e\right )\right )}}{a d f x + a c f + {\left (a d f x + a c f\right )} \cos \left (f x + e\right )^{2} + {\left (a d f x + a c f\right )} \sin \left (f x + e\right )^{2} - 2 \, {\left (a d f x + a c f\right )} \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\left (a-a\,\cos \left (e+f\,x\right )\right )\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{c \cos {\left (e + f x \right )} - c + d x \cos {\left (e + f x \right )} - d x}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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